# maximal ideals of the algebra of continuous functions on a compact set

Let $X$ be a compact^{} Hausdorff space and $C(X)$ the Banach algebra^{} of continuous functions^{} $X\u27f6\u2102$ (with the sup norm).

In this entry we are interested in identifying the maximal ideals and the character space^{} of $C(X)$. Since $C(X)$ is a Banach algebra with an identity element, there is a bijective^{} correspondence between the character space of $C(X)$ and the set of maximal ideals of this algebra, given by

$$\varphi \u27f7Ker\varphi $$ |

Hence, by identifying the character space of $C(X)$ we are able to identify its maximal ideals.

$$

Theorem 1 - Let $\mathrm{\Delta}$ be the character space of $C(X)$. For each $x\in X$ let $e{v}_{x}\in \mathrm{\Delta}$ be the point-evaluation at $x$, i.e.

$$e{v}_{x}(f)=f(x),f\in C(X)$$ |

Then the mapping $x\u27fce{v}_{x}$ is an homeomorphism between $\mathrm{\Delta}$ and $X$.

$$

Thus, the character space of $C(X)$ is homeomorphic to $X$ via point-evaluations.

Now, the maximal ideals of $C(X)$ correspond to the kernels of the point-evaluation functions. The kernel of $e{v}_{x}$, the point-evaluation at $x$, is just

$\mathrm{\{}f\in C(X):f(x)=0\}$ |

i.e., the functions that vanish at $x$.

Thus, each maximal ideal of $C(X)$ is just the set of functions that vanish in a given point.

## 0.1 Generalization to locally compact Hausdorff spaces

Now, let $X$ be a locally compact Hausdorff space^{} and ${C}_{0}(X)$ the space of continuous functions $X\u27f6\u2102$ that vanish at infinity.

There is a generalization^{} of Theorem 1 above that allows one to identify the character space of ${C}_{0}(X)$, but since this algebra is not unital unless $X$ is compact, we cannot identify its maximal ideals by the above method.

$$

Theorem 2- Let $\mathrm{\Delta}$ be the character space of ${C}_{0}(X)$. For each $x\in X$ let $e{v}_{x}\in \mathrm{\Delta}$ be the point-evaluation at $x$, i.e.

$$e{v}_{x}(f)=f(x),f\in {C}_{0}(X)$$ |

Then the mapping $x\u27fce{v}_{x}$ is an homeomorphism between $\mathrm{\Delta}$ and $X$.

$$

Thus, the character space of ${C}_{0}(X)$ is also homeomorphic to $X$ via point-evaluations.

Title | maximal ideals of the algebra of continuous functions on a compact set |
---|---|

Canonical name | MaximalIdealsOfTheAlgebraOfContinuousFunctionsOnACompactSet |

Date of creation | 2013-03-22 17:44:57 |

Last modified on | 2013-03-22 17:44:57 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46L05 |

Classification | msc 46J20 |

Classification | msc 46J10 |

Classification | msc 16W80 |

Synonym | character space of the algebra of continuous functions on a compact set |